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MEMS Inertial Sensors-Based Multi-Loop Control Enhanced by Disturbance Observation and Compensation for Fast Steering Mirror System Chao Deng 1,2,3 , Yao Mao 1,2, * and Ge Ren 1,2 1 2 3

*

Institute of Optics and Electronics, Chinese Academy of Science, Chengdu 610209, China; [email protected] (C.D.); [email protected] (G.R.) Key Laboratory of Optical Engineering, Chinese Academy of Sciences, Chengdu 610209, China University of Chinese Academy of Science, Beijing 100039, China Correspondence: [email protected]; Tel.: +86-155-200-19246

Academic Editor: Stefano Mariani Received: 26 August 2016; Accepted: 8 November 2016; Published: 15 November 2016

Abstract: In this paper, an approach to improve the disturbance suppression performance of a fast steering mirror (FSM) tracking control system based on a charge-coupled device (CCD) and micro-electro-mechanical system (MEMS) inertial sensors is proposed. The disturbance observation and compensation (DOC) control method is recommended to enhance the classical multi-loop feedback control (MFC) for line-of-sight (LOS) stabilization in the FSM system. MEMS accelerometers and gyroscopes have been used in the FSM system tentatively to implement MFC instead of fiber-optic gyroscopes (FOG) because of its smaller, lighter, cheaper features and gradually improved performance. However, the stabilization performance of FSM is still suffering a large number of mechanical resonances and time delay induced by a low CCD sampling rate, which causes insufficient error attenuation when suffering uncertain disturbances. Thus, in order to make further improvements on the stabilization performance, a cascaded MFC enhanced by DOC method is proposed. The sensitivity of this method shows the significant improvement of the conventional MFC system. Simultaneously, the analysis of stabilization accuracy is also presented. A series of comparative experimental results demonstrate the disturbance suppression performance of the FSM control system based on the MEMS inertial sensors can be effectively improved by the proposed approach. Keywords: MEMS inertial sensors; disturbance observation and compensation; multi-loop feedback control; light of sight stabilization

1. Introduction The fast steering mirrors (FSMs) play a critical role in optical fine tracking control systems, such as for adaptive optics, long-distance laser communication, line-of-sight (LOS) stabilization, which are increasingly mounted on vehicles, airplanes, spacecraft and other moving platforms [1–4]. In the classical FSM control system, the fiber-optic gyroscopes (FOGs) and charge-coupled devices (CCDs) are generally used to implement a dual closed-loop control to stabilize LOS [5,6]. High closed-loop bandwidth facilitates good closed-loop performance. However, the control bandwidth is limited mainly by mechanical resonances, time delay, and also the sensors’ noise. With the development of the MEMS industry in recent years, the performance of micro-electro-mechanical system (MEMS) inertial sensors, including accelerometers and gyroscopes, have been rapidly improved [7,8]. Considering the installation position of the sensors limited to the narrow spaces of the reverse side of the mirror, both MEMS accelerometer and gyroscope can be mounted on the frame of FSM due to the relatively small size and low weight [7–10]. Usually, the bandwidth of Sensors 2016, 16, 1920; doi:10.3390/s16111920

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Sensors 2016, 16, of 1920 2 of 12 bandwidth MEMS gyroscopes with low noise is generally less than 100 Hz, which limits the

bandwidth of the velocity closed-loop and the disturbance suppression ability; nevertheless, the bandwidth of MEMS accelerometers exceeds 800 Hz. Thus, the high bandwidth acceleration MEMS gyroscopes with low noise is generally less than 100 Hz, which limits the bandwidth of the feedback loop implemented by two linear MEMS accelerometers can be used to improve the velocity closed-loop and the disturbance suppression ability; nevertheless, the bandwidth of MEMS disturbance suppression ability of FSM control system [7,9].acceleration feedback loop implemented by accelerometers exceeds 800 Hz. Thus, the high bandwidth The acceleration feedback control (AFC) is kind of the high-precision control ability methodofto two linear MEMS accelerometers can be used to aimprove disturbancerobust suppression FSM form multi-loop feedback control (MFC), which has been widely used in some high precision control system [7,9]. systems such as telescopes, missiles and robots In FSM systems, the acceleration closed method loop The acceleration feedback control (AFC)[11–13]. is a kind of high-precision robust control improve the stiffness of the system. In previous works, some researchers on how to tocan form multi-loop feedback control (MFC), which has been widely used infocused some high precision design the acceleration controller because of the [11–13]. quadraticIndifferential in thethe FSM transfer model. systems such as telescopes, missiles and robots FSM systems, acceleration closedToloop avoid the saturation of double designed the controller as a band-pass filter and to can improve the stiffness of theintegration, system. InTang previous works, some researchers focused on how combined a CCD and accelerometers to implement dual closed-loop control [14]. Tian used a CCD, design the acceleration controller because of the quadratic differential in the FSM transfer model. accelerometers to integration, implement a Tang three-closed-loop model, a lag controller Togyroscopes avoid the and saturation of double designed thecontrol controller as and a band-pass filter and was useda CCD to accomplish acceleration control, which the disturbance suppression combined and accelerometers to implement dual improved closed-loop control [14]. Tian used a CCD, gyroscopes and accelerometers a three-closed-loop model, andnot a lag controller performance powerfully [9,15]. to Butimplement the error rejection ability of the control FSM system is still adequate was used to accomplish acceleration control, which improved the disturbance suppression performance when suffering some uncertain disturbances, because of the mechanical resonances, CCD time delay powerfully [9,15]. the error rejection ability of the FSM system is still not adequate when suffering and the drift noiseBut in MEMS inertial sensors [14]. some uncertain disturbances, thestabilization mechanical performance, resonances, CCD time delay and the In this paper, to furtherbecause enhanceofthe we proposed a new FSMdrift noise in MEMS inertial sensors [14]. stabilization control method, which combines the MFC with the disturbance observation and In this paper, to method. further The enhance performance, we of proposed newthe FSM compensation (DOC) MFC the can stabilization be used to improve the stiffness the FSM,a and stabilization control method, which combines the MFC with the disturbance observation DOC is utilized to estimate and compensate the outer base disturbance [16–20]. The compensatingand compensation (DOC) method. The on MFC be used to model improve the controlled stiffness ofplant the FSM, andInthe DOC precision of DOC mostly depends thecan mathematical of the [16,17]. FSM iscontrol utilizedsystems, to estimate and compensate the outer base disturbance [16–20]. The compensating precision the acceleration response mathematical model can be achieved in high accuracy by of DOC mostly depends on the mathematical model of the controlled plant [16,17]. In FSM control a spectral analyzer, which is conducive to DOC fulfillment in practical systems. As a consequence, a systems, the acceleration response mathematical model can be achieved in high accuracy by a spectral detailed introduction to the model of the FSM is presented in Section 2. And then the theory analysis analyzer, which is conducive to DOC fulfillment in practical systems. As a consequence, a detailed and the controller design are shown in Sections 3 and 4. The verification experiment is discussed in introduction to the model of the FSM is presented in Section 2. And then the theory analysis and the Section 5. Concluding remarks are presented in Section 6. controller design are shown in Sections 3 and 4. The verification experiment is discussed in Section 5. Concluding remarks are presented in Section 6. 2. FSM System Control Model

2. FSM Control Model A System FSM is generally defined as a mirror mounted to a flexure support system and driven by actuators [21,22]. The schematic of the FSM control system is illustrated in Figure 1. The by A FSM is generally definedstructure as a mirror mounted to a flexure support system and driven voice coil[21,22]. motors are used to drive the FSM platform achieve the stabilization of LOS [22].in Figure 1. actuators The schematic structure of the to FSM control system is illustrated The voice coil motors are used to drive the FSM platform to achieve the stabilization of LOS [22]. CCD

Light Source

θ

Sensors

θref

Motion Controller

Mirror

Voice Coil Motor

Driver

Disturbance

Figure 1. The structure of the fast steering mirror (FSM) control system. Figure 1. The structure of the fast steering mirror (FSM) control system.

The mechanical part of the FSM is a typical resonance element and the voice coil motors are Thefirst-order mechanicalinertial part of element the FSM is typical resonance element the voice coil motors are a a typical inathe mathematical model. and Therefore, the FSM acceleration typical first-order inertial element in the mathematical model. Therefore, the FSM acceleration controlled model including a quadratic differential can be expressed as follows [9,15,21,23]: controlled model including a quadratic differential can be expressed as follows [9,15,21,23]: ..

Ga ( s ) =

θ (s) = U (s)

s2 2ξ +$ s + 1 Te s + 1 n K

s2

$n

2

·

The control strategy of FSM goes into particulars as follows.

(1)

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Ga (s) 

 (s) U(s)



s

2



K 2

 n2  n

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 s1

s2 Te s  1

(1) 3 of 12

The control strategy of FSM goes into particulars as follows.

2.1. Multi-Loop Feedback Control (MFC) 2.1. Multi-Loop Feedback Control (MFC) The classical stabilization control structure of multi-loop feedbackcontrol controlin inFSM FSM system system is is shown The classical stabilization control structure of multi-loop feedback shown in Figure 2 [9,15]. in Figure 2 [9,15]. d s2

 ref

Cp (s)

wref

w

a

aref

Cv ( s )

Ga ( s )

Ca ( s )

1/ s

1/ s



MEMS Accelerometers MEMS Gyroscope

CCD

s) iscontrolled 2. Classical multi-loop feedback structure. the controlled Figure 2.Figure Classical multi-loop feedback controlcontrol (MFC)(MFC) structure. Ga (s)Gisa (the plant, Ca (s) C ( s ) C ( s ) C ( s ) plant, is the acceleration controller, is the velocity controller, is theθd is the is the acceleration controller, Cv (s) is the velocity controller, C p (s) is the position controller, p a v outer disturbance angle ,andθre is the outer given disturbance target position. position controller, angle ,and  ref is the given target f isthe d

position.

According to the control structure, it is easy to obtain, According to the control structure, it is easy to obtain,

C p Cv Ca Ga s12

1 1 θre f + θd (2) 2 1 1 1 + Ca Ga + Cv Ca Ga s + C p Cs v Ca Ga s2   1 + Ca Ga + Cv Ca Ga 1s + C p Cv Ca Ga s12(2) ref d 1 1 1 1 1  CaGa  CvCaGa +CpCvCaGa 2 1  CaGa  CvCaGa  CpCvCaGa 2 s s s s As the purpose of this work is not the tracking problem, but the stabilization of FSM, the given θ=

Cp1CvCaGa

target position θrepurpose The of FSM with the classic three As the thisdisturbance work is not thesuppression tracking problem, but the stabilization of FSM, theclosed-loops given f = 0. of ref  0 . (3): position The disturbance suppression of FSM with the classic three closed-loops methodtarget is given in Equation method is given in Equation (3):

E MFC =

θ

1

=

θd  EMFC

(3)

1 1+ Ca Ga + Cv Ca Ga 1s + C p Cv Ca Ga s12 1 1 d 1  CaGa  CvCaGa  CpCvCaGa 2 s s

(3)

Therefore, it can be seen that the performance of MFC depends on the entire three closed-loops. Therefore, it can be seen that theobtain performance of MFC depends the entire three closed-loops. An appropriate controllers design can the minimum E MFC , on which means the least effect from An appropriate controllers design can obtain the minimum , which means the least effect E MFC the disturbance θd . from the disturbance d .

2.2. Multi-Loop Feedback Control with Disturbance Observation and Compensation (MFC-DOC) 2.2. Multi-Loop Feedback Sensors 2016, 16, 1920

Control with Disturbance Observation and Compensation (MFC-DOC)

The modified structure of MFC-DOC is shown in Figure 3.

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The modified structure of MFC-DOC is shown in Figure 3. d s

 ref

wref

Cp (s)

aref

Cv ( s )

u

Ca ( s )

2

w

a

Ga ( s )

1/ s

-

1/ s



+

Ga ( s )

-

C f (s) MEMS Accelerometers MEMS Gyroscope CCD

Figure 3.Figure The multi-loop feedbackfeedback control with disturbance observation and compensation (MFC-DOC) 3. The multi-loop control with disturbance observation and compensation e Ga ( s) ismodel (MFC-DOC) structure. the approximate model plant, of the which controlled which can be structure. Ga (s) is the approximate of the controlled canplant, be obtained by fitting the measured transfer andmeasured C f (s) is transfer the disturbance compensation C f ( s) is the controller. obtained by model, fitting the model, and disturbance compensation controller.

The closed-loop acceleration is given in Equation (4):

a  uGa  s2d u  (aref  a)Ca  (a  uGa )C f

(4)

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The closed-loop acceleration is given in Equation (4): a = uGa + s2 θd ea )C f u = ( are f − a)Ca − ( a − u G

(4)

As the element u is a substitutable factor, after substitution, it is easy to obtain: ea C f ] a − s2 θd = ( are f − a)Ca Ga − [ aC f Ga − ( a − s2 θd ) G

(5)

ea )C f ] a = Ca Ga are f + (1 − G ea C f )s2 θd [ 1 + Ca Ga + ( Ga − G

(6)

So, a=

Ca Ga ea )C f 1 + Ca Ga + ( Ga − G

are f +

ea C f )s2 (1 − G ea )C f 1 + Ca Ga + ( Ga − G

(7)

θd

According to the control structure, it is easy to get:  are f = Cv wre f − w)

(8)

Then, the closed-loop velocity transfer function can be given: w=

ea C f )s (1 − G Cv Ca Ga 1s wre f + θ 1 ea )C f + Cv Ca Ga ea )C f + Cv Ca Ga 1 d 1 + Ca Ga + ( Ga − G 1 + Ca Ga + ( Ga − G s s

(9)

Similarly, the closed-loop angular position will be depicted as follows: θ=

C p Cv Ca Ga

1 s2

ea )C f +Cv Ca Ga 1 +C p Cv Ca Ga 1+Ca Ga +( Ga − G s

1 s2

θre f +

ea C f ) (1− G ea )C f +Cv Ca Ga 1 +C p Cv Ca Ga 1+Ca Ga +( Ga − G s

1 s2

θd

(10)

As θre f = 0, the disturbance suppression of the system with MFC-DOC method can be given as follows: E MFC− DOC =

ea C f 1−G θ = ea )C f + Cv Ca Ga 1 + C p Cv Ca Ga 12 θd 1 + Ca Ga + ( Ga − G s s

(11)

Hence, the performance of MFC-DOC depends on the entire three loops controllers as well as the DOC structure. 3. Performance Analysis From the above derivation, the disturbance rejection characteristics of the MFC and the MFC enhanced by DOC are given clearly. The disturbance rejection characteristics of the MFC can be presented as follows: E MFC =

1 1 + Ca Ga + Cv Ca Ga 1s + C p Cv Ca Ga s12

(12)

And the disturbance rejection characteristics of the MFC-DOC can be presented as follows: E MFC− DOC =

ea C f 1−G ea )C f + Cv Ca Ga 1 + C p Cv Ca Ga 12 1 + Ca Ga + ( Ga − G s s

(13)

ea is the approximate model of the controlled plant, where Ga is the controlled plant, while G which can be obtained by fitting the measured acceleration transfer function. By a spectral analyzer,

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the acceleration response of the FSM can be obtained in a high accuracy. Thus, we can easily get ea ≈ Ga , and it is clear that: G ea )C f ≈ 0 ( Ga − G (14) So, the denominators of E MFC and E MFC− DOC are almost equal, moreover, the denominators show the stability of the system, which means that the introduction of DOC method has an almost minimal effect on the stability of system. Spontaneously, when comparing with MFC, the disturbance rejection characteristics of MFC-DOC can be simplified as follows: ea C f Eˆ MFC− DOC = 1 − G

(15)

According to Equation (15), the disturbance compensation controller C f is of great concern for the effect of DOC, and a zero error remain will be achieved in theory if the controller is designed to an ideal controller as follows: ea −1 Cf ∼ (16) =G Eˆ MFC− DOC → 0

(17)

ea , which can compensate disturbances The ideal DOC controller is the inverse transfer function of G in all frequencies in theory. As depicted in Figure 3, the input of C f is the estimate of disturbance force and the purpose of C f is to convert the estimated disturbance force into the driver input, by which the Sensors 2016, 16, 1920 6 of 13 ea , the ability of disturbance compensation disturbance compensation can be completed. After getting G depends primarily on the C f design. The ideal DOC controller of FSM can be presented as follows: 2 s1 2ξn  + $n s + 1 Te sT2e s1+ 1

s2



2 s2 C f _ ideal  n 2 n

C f _ideal =

$

(18)

·s

K

Magnitude

(18) K s2 The open loop natural frequency of FSM  n is approximately between several Hz to tens of The open loop natural frequency of FSM $n is approximately between several Hz to tens of Hz, and the damping factor  is much smaller than 1 [1,21]. Due to the high sampling frequency Hz, and the damping factor ξ is much smaller than 1 [1,21]. Due to the high sampling frequency in FSM system, the electrical lag factor Te is also much smaller than 1. Thus, it is easy to depict the in FSM system, the electrical lag factor Te is also much smaller than 1. Thus, it is easy to depict the diagrammatic sketch of the ideal DOC controllerin in frequency frequency domain in Figure 4. 4. diagrammatic sketch of the ideal DOC controller domainasasshown shown in Figure

-40dB/dec

n

Low Frq

1 Te

Middle Frq

20dB/dec

High Frq

Figure 4. The diagrammatic sketch observation and compensation (DOC) Figure 4. The diagrammatic sketchofofthe theideal ideal disturbance disturbance observation and compensation (DOC) controller in frequency domain. controller in frequency domain.

The full frequency range is divided into three parts in Figure 4, including low frequency,

The full frequency range is divided into three parts in Figure 4, 2 including low frequency, middle frequency and high frequency. Due to the double integration and the second-order s the middle frequency and high frequency. Due to the double integration s2 and second-order element, element, the DOC controller decreases in a slope to −40 dB at low frequency and remains constant the DOC controller decreases in a slope to −40 dB at low frequency and remains constant at middle at middle frequency. Because of the Te s1 element, it rises in a slope to 20 dB at high frequency. frequency. Because of the Te s + 1 element, it rises in a slope to 20 dB at high frequency. However, However, because of the of lowaccelerometers sensitivity of accelerometers at low the right estimatedforce because of the low sensitivity at low frequency, the frequency, right estimated disturbance disturbance force is partly submerged in the noise, which causes the DOC controller to become is partly submerged in the noise, which causes the DOC controller to become saturated as a result of saturated as a result of the quadratic integration influence. At high frequency, the disturbance effect the quadratic integration influence. At high frequency, the disturbance effect is mostly restrained by is mostly restrained by the passive disturbance suppression of FSM platform. As the DOC the passive disturbance suppression of FSM platform. As the DOC controller can be regarded as a controller can be regarded as a proportional controller at middle frequency, it is easy to realize it by proportional controller at middle frequency,ofitthe is easy to frequency, realize it by low-pass filter.is Thus, to take a low-pass filter. Thus, to take advantage middle theaDOC controller designed advantage of the middle frequency, the DOC controller is designed in Equation (19): in Equation (19): Cf 

K f (Te s1) s  2 f w f s  w f 2 2

(19)

where Te s1 is used to compensate phase lag. The second-order resonance element is used to filter the high-frequency noise of accelerometers and compensate the quadratic integration partly.

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Cf =

K f ( Te s + 1) s2

(19)

+ 2ξ f w f s + w f 2

where Te s + 1 is used to compensate phase lag. The second-order resonance element is used to filter the high-frequency noise of accelerometers and compensate the quadratic integration partly. Substituting Equations (1) and (19) into Equation (15), the disturbance rejection transfer function of MFC-DOC is presented as follows: ea C f = 1 − Eˆ MFC− DOC = 1 − G

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ξ s4 +2 + n2w f 2 n2w f f ξ s4 +2 n2w f 2 n2w f

$ The damping = factor 

(s2 +2ξ

!

ξf

1 + n2

s3 +(

KK f $n 2 s2 $n s+$n 2 )(s2 +2ξ f w f s+w f 2 )

KK f + 2− 2 wf wf nw f

4ξξ f

1

) s2 +2( ξ

7 of 13

ξf +w n f

) s +1 (20) $ $ $ $n w f 2 $ is much smaller ! than 1. The designed value of w f should be much ξf

1

3

4ξξ f

1

2

ξ

ξf

+ 2 s +( $n 2 + $n w + w 2 ) s +2( $n + w f ) s +1 $ $ $n wof f bigger than  n , otherwise the bandwidth the controllerf willf be too low. Thus, the Eˆ MFC  DOC can

be simplified as follows: The damping factor ξ f is much smaller than 1. The designed value of w f should be much bigger ˆ than $n , otherwise the bandwidth of the 1controller KK f 2will be  too flow. Thus, the EMFC− DOC can be (  ) s  2(  ) s  1 simplified as follows: n wf  n2 w f 2 Eˆ MFC  DOC  ( 1 2 − KK2f )s2 + 2( ξ + ξ f )s + 1 (21) 2 $n s w f $f n w f  ˆ E MFC− DOC ≈ (21)  2(  ξ )s  1 s2n 22 + 2($nξ +wwf f )s + 1 $n n f In Equation into a Equation (21), (21), the thedisturbance disturbancerejection rejectiontransfer transferfunction functionofofMFC-DOC MFC-DOCcan canbebedesigned designed into lead-lag corrector form, thethe diagrammatic sketch of of which is is depicted in in Figure 5. 5. a lead-lag corrector form, diagrammatic sketch which depicted Figure Magnitude

n 1  KK f n 2 / w f 2

n

-20dB/dec

Figure The diagrammatic diagrammatic sketch sketch of Figure 5. 5. The of the the lead-lag lead-lag corrector. corrector.

As at at low frequency could be neglected because of the As is is shown shownin inFigure Figure5,5,the thecompensation compensation low frequency could be neglected because of low the sensitivity of accelerometers at low But when the FSM system is suffering middle or high low sensitivity of accelerometers at frequency. low frequency. But when the FSM system is suffering middle or frequency disturbance, it substantially improves the disturbance rejection ability. So without doubt, high frequency disturbance, it substantially improves the disturbance rejection ability. So without there be:will be: doubt,will there E MFC− DOC < E MFC  1 (22) EMFC  DOC  EMFC 1 (22) It indicates that depending on the lead-lag controller design, the MFC-DOC method has a stronger It indicates that ability depending on the controller design, the the MFC-DOC method disturbance rejection than the onlylead-lag MFC method; in other words, system will sufferhas lessa stronger disturbance rejection ability than the only MFC method; in other words, the system will influence from the outer disturbance and the remainder error of LOS reduces further. suffer lessMFC-DOC, influence from the outer disturbance remainder error of LOS reduces further. With the sensitivity function of and FSMthe becomes: With MFC-DOC, the sensitivity function of FSM becomes: !. ! ( Ga +∆Ga )Ca Cv

1

Ga Ca Cv

1

Ga Ca Cv

1

· − · · 1+( Ga +  (G∆G a )CGa +[( )CGCa +∆Ga )−Gea ]C f 1s 1+Ga CaG+( CGCa −Gea )C f s 1   1+Ga CGa +( CGCa −Gea )C f s1  a a a v a a v a a v S MFC− DOC =   ∆Ga /G  /     a  1  (Ga  Ga )Ca  [(1−   1  GaCa  (Ga  Ga )C f s  s s G   G )  G ] C 1  G C  ( G  G ) C e G C a a a f a a a a f a    f SMFC  DOC  = ea ]C f 1+( Ga +∆Ga )Ca +[( Ga +∆Ga )− G Ga / Ga ea C f 1− G ≈ 1  GaCGef )C 1+ Ga Ca +( Ga − a f  e 1 − G C 1≈  (Ga  aGaf)Ca  [(Ga  Ga )  Ga ]C f 1 + Ga Ca 1  GaC f  1  GaCa  (Ga  Ga )C f



1  GaC f 1  GaCa

(23)

(23)

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According to the previous demonstration, if the DOC controller is designed as Equation (19), S MFC− DOC < S MFC can be obtained [9], which means that the introduction of DOC can also improve Sensors 2016, 16, 1920 8 of 13 the robustness of the traditional MFC system. If the controlled model of the system changes Sensors 2016, 16, 1920 8 of 13 substantially, the stability of the velocity loop with DOC will suffer less than that with MFC. 4. Controller Design 4. Design 4. Controller Controller Design The opened loop characteristic of FSM acceleration response measured by spectral analyzer The opened loop FSM acceleration response measured by spectral analyzer fromThe 1 Hz to 1000loop Hz characteristic ischaracteristic shown in Figure 6. acceleration opened ofof FSM response measured by spectral analyzer from 1 Hz toHz 1000 is shown in Figure 6. 1from Hz to 1000 is Hz shown in Figure 6. FSM acceleration response

Phase(Deg) Phase(Deg)

Magnitude(dB) Magnitude(dB)

20

FSM acceleration response

20 10 10 0 0 -10 -10 -20 -20 -30 0 10 -30 0 10 0

1

Measure result Fit curve result Measure result 3 Fit curve result10

2

10

10

1

2

10

3

10

10

0 -100 -100 -200 -200 -300 -300 -400 -400 -500 0 10 -500 0 10

1

2

10

1

10

3

10

Frequency(Hz)

10

2

3

10

10

Frequency(Hz)

Figure 6. The opened loop characteristic of the FSM acceleration response. Figure acceleration response. response. Figure 6. 6. The The opened opened loop loop characteristic characteristic of of the the FSM FSM acceleration

Because of the high bandwidth of the MEMS accelerometer KXR94-2050 (Kionix, New York, Because of the high bandwidth of the MEMS accelerometer (Kionix, New York, NY, Because USA) [24], thehigh frequency characteristic canaccelerometer be measuredKXR94-2050 toKXR94-2050 1000 Hz. By theNew curve-fitting of the bandwidth of the MEMS (Kionix, York, NY, NY, USA) [24], the frequency characteristic can be measured to 1000 Hz. By the curve-fitting method and transfer function identification, the mathematic transfer function model of FSM USA) [24], the frequency characteristic can be measured to 1000 Hz. By the curve-fittingthe method method and transfer function identification, the mathematic transfer function model of the FSM acceleration can be obtained follows: transfer function model of the FSM acceleration and transfer response function identification, theasmathematic acceleration can as befollows: obtained as follows: response canresponse be obtained 0.00225 s2 Ga   (24) 2 0.00225 s2 s2  1 s  0.019s  1  0.0005 0.00225 s Ga  0.00078 (24) 2 e (24) Ga = 0.000782s  0.019s  1 ·0.0005s  1 + 1  is about 6 Hz and the 0.00078s 0.019s + 1of 0.0005s It is obvious that the open loop natural+frequency FSM system n  n DOC It is obvious that about the open loop natural frequency of FSManalysis, system the is about 6 Hz and damping factor that According to theofprevious canthe be It is obvious open0.057. loop natural frequency FSM system $n is about 6 Hzcontroller and the damping  isthe damping factor is about 0.057. According to the previous analysis, the DOC controller can be  factor ξ is about 0.057. According to the previous analysis, the DOC controller can be designed designed as Equation (23). The natural frequency w f , which decides the bandwidth of the filter as is Equation The natural w ffrequency , which decides the bandwidth of bandwidth the filter is chosen by the w f , which designed (23). as Equation (23). frequency The natural decides the of the filter is chosen by the characteristics the sensors noise. The bandwidth of the accelerometer is above 800 the Hz characteristics of the sensorsofnoise. The bandwidth of the accelerometer is above 800 Hz and chosen by the characteristics of the sensors noise. The bandwidth of the accelerometer is above 800 Hz and the characteristics of the accelerometer noise are depicted in Figure 7. characteristics of the accelerometer noise are depicted in Figure 7. and the characteristics of the accelerometer noise are depicted in Figure 7. Accelerometer Noise Accelerometer Noise

40 30 30 20 20 10 10 0 0 -10 -10 -20 -20 -30 -30 -40 -40

0

2

0

2

4

6

Time(s) 4 6 Time(s)

Spectrum of Noise Spectrum of Noise

3 2 2 Amplitude(arcsecond/s ) ) Amplitude(arcsecond/s

2 2 Amplitude(arcsecond/s ) ) Amplitude(arcsecond/s

40

8

8

3 2.5

X: 299.9 Y: 2.161 X: 299.9 Y: 2.161

2.5 2 2 1.5 1.5 1

X: 200 Y: 0.6666 X: 200 Y: 0.6666

1 0.5 0.5 0 0

0

0

500

1000

1500

Frequency 500 1000 (Hz) 1500 Frequency (Hz)

2000

2000

Figure 7. MEMS accelerometer noise characteristics. Figure 7. MEMS accelerometer noise characteristics.

The peak value of the MEMS accelerometer transformed noise is equal to 0.0097°/s2, and the 2. Besides The peak value0.0022°/s of the MEMS accelerometer transformed noise is equal 0.0097°/s2, and the RMS value is about some “spikes”, the amplitude value of to amplitude-frequency 2 RMS is about . Besides some high “spikes”, the amplitude valuecould of amplitude-frequency curvevalue is smooth and0.0022°/s the “spikes” are some frequency noise, which be filter out by the curve is smooth and the “spikes” are some high frequency noise, which could be filter out by the

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The peak value of the MEMS accelerometer transformed noise is equal to 0.0097◦ /s2 , and the ◦ 2 RMSSensors value2016, is about 16, 1920 0.0022 /s . Besides some “spikes”, the amplitude value of amplitude-frequency 9 of 13 curve is smooth and the “spikes” are some high frequency noise, which could be filter out by the DOC DOC controller. Considering andofthe lag the of filter, the bandwidth of DOC controller is at controller. Considering the noisethe andnoise the lag filter, bandwidth of DOC controller is designed designed at 40 Hz. Figure 8 shows the characteristics of the MEMS gyroscope noise. 40 Hz. Figure 8 shows the characteristics of the MEMS gyroscope noise. Gyroscope Noise

Spectrum of Noise 8

Amplitude(arcsecond/s)

Amplitude(arcsecond/s)

300 200 100 0 -100 -200 -300

0

2

4

6

8

7 6 5 4 3 2 1 0

0

Time(s)

500

1000

1500

2000

Frequency (Hz)

Figure 8. Micro-electro-mechanicalsystem system (MEMS) (MEMS) gyroscope Figure 8. Micro-electro-mechanical gyroscopenoise noisecharacteristics. characteristics.

The bandwidth of the MEMS gyroscope SCR1100-D02 (Murata Manufacturing, Nagaokakyo, The bandwidth of the MEMS gyroscope SCR1100-D02 (Murata Manufacturing, Nagaokakyo, Kyoto) [25] is less than 100 Hz. The noise mainly focuses on the frequencies ranging from 0 to 200 Hz. Kyoto) [25] is less than 100 Hz. The noise mainly focuses on the frequencies ranging from 0 to 200 Hz. 2, and the RMS value is about The peak value of the MEMS gyroscope noise is equal to 0.069°/s ◦ /s2 , and the RMS value is about The 0.016°/s peak value of the MEMS gyroscope noise is equal to 0.069 2, which satisfy the requirement of the control system. 0.016◦ /s2Thus, , which the requirement of the control system. thesatisfy DOC controller can be designed as follows:

Thus, the DOC controller can be designed as follows: Cf 

27161(0.0005s1)

(25)

27161 × (0.0005s + 1) s2  355.4 s  63165 Cf = 2 (25) s + 355.4s + 63165 where the damping factor  f  0.707 and the natural frequency w f  40 Hz  80π rad/s , which where damping factor ξf = 0.707 the natural w f =control 40 Hz in = acceleration 80π rad/s, which arethe both configured by the noise of and sensors. To avoidfrequency the inaccurate closed are bothloop, configured by the noise of sensors. To avoid the inaccurate control in acceleration closed loop, the acceleration controller is designed as follows [15]: the acceleration controller is designed as follows [15]: 260 0.0011s  1  260s 0.00092 0.0011ss +11

Ca 

Ca =

×

(26)

(26)

s the0.00092s 1 FSM has been simplified. Therefore, With the completion of loop acceleration, model of+the the traditional PI controller can meet the velocity closed-loop and also the position closed-loop [15].

With the completion of loop acceleration, the model of the FSM has been simplified. Therefore, the traditional PI controller can meet the velocity closed-loop and also the position closed-loop [15]. 5. Experimental Verification As shownVerification in Figure 9a, an apparatus constructed by two superimposed FSM systems is used to 5. Experimental test and verify the previous analysis. One is used to stabilize the LOS; the other is used to simulate As shown inThe Figure 9a, anplatform apparatus constructed bydisturbance two superimposed systems is used disturbance. stabilized is mounted on the platform FSM and the laser light is to test and verify the previous analysis. One is used to stabilize the LOS; the other is used to simulate fastened on the top of stabilized platform directly as a reference of LOS. Both of the platforms are disturbance. stabilized platform mounted platformare and thetolaser light is driven by The the voice coil motors. Theiseddy sensorson in the the disturbance disturbance platform used measure fastened on the topinput of stabilized platform directly reference oftoLOS. Both the platforms the disturbance of the stabilized platform andas thea CCD is used obtain the of stabilization error.are driven byMEMS the voice coil motors. The and eddy sensors the disturbance platform are used to measure Four linear accelerometers one MEMSingyroscope are set on the stabilized platform to the disturbance of acceleration the stabilized platform the CCD is used to the stabilization error. measure the input angular and angular and velocity, respectively. In obtain the experiment, all of the FourMEMS MEMSinertial linearsensors accelerometers and one MEMS gyroscope are set on the stabilized platform and the disturbance eddies are at 5000 Hz sampling frequency, moreover, to measure the angular acceleration andfrequency angular velocity, respectively. In time the experiment, all of the MEMS the CCD has only 100 Hz working and 20 ms (two frames) delay in addition. inertial sensors and the disturbance at 5000 Hz sampling moreover, thecontrol CCD has The disturbance suppressioneddies of FSMare system is measured whenfrequency, the stabilized platform closed and the disturbance platform opened. Then, four experiments in different frequency onlyis100 Hz working frequency and 20 msis(two frames) time delay in addition. disturbance are compared as follows, andsystem the error of MFC and MFC-DOC are given in Figurecontrol 10 The disturbance suppression of FSM is measured when the stabilized platform and Table is closed and 1. the disturbance platform is opened. Then, four experiments in different frequency

disturbance are compared as follows, and the error of MFC and MFC-DOC are given in Figure 10 and Table 1.

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9 ofof12 10 13 10 of 13

MEMS accelerometers MEMS accelerometers

MEMS gyroscope MEMS gyroscope

CCD CCD

Laser light Laser light

Stabilized platform Stabilized platform Flexure pivot Flexure pivot

Voice coil Voice motor coil motor

stabilization error stabilization error

Stabilized platform base Stabilized platform base

Disturbance platform Disturbance platform Disturbance Disturbance

Flexure pivot Flexure pivot

Eddy Eddy

Disturbance platform base Disturbance platform base

(a) (a)

(b) (b) Figure 9. Experimental apparatus. (a) Principle of apparatus; and (b) Prototype of apparatus. Figure 9. 9. Experimental Experimentalapparatus. apparatus. (a) (a)Principle Principleof ofapparatus; apparatus;and and(b) (b) Prototype Prototype of of apparatus. apparatus. Figure Error at 1Hz Error at 1Hz 10 10

Error(arcsecond) Error(arcsecond)

Error at 8Hz Error at 8Hz

MFC only MFC only MFC-DOC MFC-DOC

10 10

5

Error at 20Hz Error at 20Hz

MFC only MFC only MFC-DOC MFC-DOC

10 10

5

5

0

0

-5

5

5

5

0

0

0

0 -5

0 -5

-5

-5

-5

-5

-5

-10 -10

-10 -10

-10 -10

-10 -10

0 0

1 1 Time(s) Time(s)

(a) (a)

2 2

0 0

1 1 Time(s) Time(s)

(b) (b)

2 2

MFC only MFC only MFC-DOC MFC-DOC

10 10

5

5

0

Error at 40Hz Error at 40Hz

MFC only MFC only MFC-DOC MFC-DOC

0 0

1 1 Time(s) Time(s)

(c) (c)

2 2

0 0

1 1 Time(s) Time(s)

2 2

(d) (d)

Figure 10. (a) Line-of-sight (LOS) error at 1 Hz disturbance; (b) LOS error at 8 Hz disturbance; Figure 10. (a) (a) Line-of-sight(LOS) (LOS)error error 1 Hz disturbance;LOS (b) error LOS error atdisturbance; 8 Hz disturbance; Figure at(d) 1atHz disturbance; at 8 Hz (c) LOS (c) LOS10. errorLine-of-sight at 20 Hz disturbance; and LOS error at 40(b) Hz disturbance. (c) LOS error at 20 Hz disturbance; and (d) LOS error at 40 Hz disturbance. error at 20 Hz disturbance; and (d) LOS error at 40 Hz disturbance.

It is obvious that the disturbance suppression performance of MFC-DOC is almost equal to the It is obvious that the disturbance suppression performance of MFC-DOC is almost equal to the MFCIt method at 1that Hz;the however, it is suppression improved heavily at 8 Hzofand 20 Hz. Due to theequal passive is obvious disturbance performance MFC-DOC is almost to MFC method at 1 Hz; however, it is improved heavily at 8 Hz and 20 Hz. Due to the passive disturbance suppression the FSM heavily platform, disturbance effect mostly the MFC method at 1 Hz;characteristics however, it isof improved atthe 8 Hz and 20 Hz. Duehas to been the passive disturbance suppression characteristics of the FSM platform, the disturbance effect has been mostly restrained, thus, it is the improvement at above 40 Hz is unapparent. restrained, thus, it is the improvement at above 40 Hz is unapparent.

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disturbance suppression characteristics of the FSM platform, the disturbance effect has been mostly restrained, thus, it is the improvement at above 40 Hz is unapparent.

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Table 1. The detailed circumstances of stabilization error comparison. Table 1. The detailed circumstances of stabilization error comparison. RMS 1 (”)RMS 1 (’’) Disturbance Disturbance Frequency Frequency MFC MFC-DOC MFC MFC-DOC 1 Hz 1.0252 1.0241 1 Hz 1.0252 1.0241 8 Hz 5.9126 1.4516 8 Hz 5.9126 1.4516 20 Hz 20 Hz3.0473 3.0473 1.4623 1.4623 40 Hz 1.4017 1.0454 40 Hz 1.4017 1.0454 1

MaxMax PeakPeak (’’) (”) MFC MFC-DOC MFC-DOC MFC 4.8598 5.1072 4.8598 5.1072 12.2017 4.3802 4.3802 12.2017 7.6593 7.6593 3.7520 3.7520 4.7773 4.7773 4.5171 4.5171

1

RMS standsfor forthe theRoot Root Mean thethe error. RMS stands MeanSquare Squareofof error.

Simultaneously, the total disturbance suppression characteristic of the two methods, shown in Simultaneously, the total disturbance suppression characteristic of the two methods, shown in Figure 11, can be obtained. It can be clearly seen that the disturbance suppression of the FSM Figure 11, can be obtained. It can be clearly seen that the disturbance suppression of the FSM system system has been improved heavily by DOC at middle frequency, while it almost has no effect at low has been improved heavily by DOC at middle frequency, while it almost has no effect at low frequency, frequency, which proves the previous analysis. which proves the previous analysis. Distuibance Suppression 0 MFC only MFC-DOC

Magnitude(dB)

-10

-20

-30

-40

-50

-60 0 10

1

10

2

10

Frequency(Hz)

Figure 11. Disturbance suppression characteristics.

6. 6. Conclusions Conclusions The disturbance rejection of LOS is the main purpose of the stabilization of the FSM control system. MEMS MEMS inertial inertial sensors sensors have have been been tried tried to to use use in the FSM system because of their relatively small size, size,low low weight low power consumption. order totheenhance the performance, stabilization weight and and low power consumption. In order In to enhance stabilization performance, the MFC is recommended to raise of thethe stiffness of the FSM system and make stronger the MFC is recommended to raise the stiffness FSM system and make stronger disturbance disturbance rejection. toimprovement make furtherofimprovement of the stabilization rejection. In this paper,In to this makepaper, further the stabilization performance, theperformance, optimization the optimization MFC withThe DOC is proposed. The structure of thisand method is discussed for MFC with DOCfor is proposed. structure of this method is discussed compared with onlyand the compared withwhich only has the proved MFC method, which has provedSimultaneously, its advantages the in theory. MFC method, its advantages in theory. analysisSimultaneously, of stabilization the analysis of method stabilization accuracy of this method is presented by of contrasting experiments. A accuracy of this is presented by contrasting experiments. A series comparative experimental series comparative experimental results show that the disturbance performance of resultsof show that the disturbance suppression performance of the FSM suppression control system based on the the FSM control system on the MEMS inertial sensors can method. be effectively improved by the MEMS inertial sensors canbased be effectively improved by the proposed proposed method. Future work will concentrate on improving the error attenuation performance of the LOS at Future work will concentrate on improving error attenuation performance of the LOS at low frequency. The use of gyroscopes may be anthe effective method to estimate the low frequency disturbance, which next work. fusionmethod of accelerometers can low frequency. The will use be of our gyroscopes mayThe be sensor an effective to estimateand the gyroscopes low frequency also be underwhich review. disturbance, will be our next work. The sensor fusion of accelerometers and gyroscopes can also be under review. Acknowledgments: Comments and suggestions from T. Tang and J. Tian during this study are very much appreciated.

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Acknowledgments: Comments and suggestions from T. Tang and J. Tian during this study are very much appreciated. Author Contributions: Chao Deng is the head of the research group that conducted this study. He contributed to research with his original idea and he carried out the experiment, and writing this paper. Yao Mao joined analyzing the theory and writing this paper. Ge Ren contributed to the research through his general guidance and advice. Conflicts of Interest: The authors declare no conflict of interest.

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